/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 Derive Milne's method by applyin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 15

Derive Milne's method by applying the open Newton-Cotes formula (4.29) to the integral $$ y\left(t_{i+1}\right)-y\left(t_{i-3}\right)=\int_{t_{i-3}}^{t_{i+1}} f(t, y(t)) d t. $$

Short Answer

Expert verified
The final form of Milne's method, derived using the open Newton-Cotes formula, is \[y(t_{i+1}) \approx y(t_{i-3})+\frac{h}{3}[f(t_{i-3}, y(t_{i-3}))+4f(t_{i-2}, y(t_{i-2}))+2f(t_{i-1}, y(t_{i-1}))+4f(t_i, y(t_i))+f(t_{i+1}, y(t_{i+1}))].\]

Step by step solution

01

Understand the Integral Problem

The integral represents the difference of the function \(y(t)\) over the range [\(t_{i-3}\), \(t_{i+1}\)]. The function to integrate, \(f(t, y(t))\), represents the derivative of \(y(t)\)
02

Apply Newton-Cotes Formula

The 4th order Newton-Cotes formula, a four-point formula, is given by \[\int_{a}^{e} f(x) d x \approx \frac{h}{3}\left[f(a)+4 f(b)+2 f(c)+4 f(d)+f(e)\right].\]Replace \(a\) with \(t_{i-3}\), \(b\) with \(t_{i-2}\), \(c\) with \(t_{i-1}\), \(d\) with \(t_i\) and \(e\) with \(t_{i+1}\), and \(h\) with \(t_{i}-t_{i-1}\) (assuming uniform steps).
03

Formulate the Milne's Method

By substituting the 4th order Newton-Cote's formula above, we arrive at \[y(t_{i+1})-y(t_{i-3}) \approx \frac{h}{3}[f(t_{i-3}, y(t_{i-3}))+4f(t_{i-2}, y(t_{i-2}))+2f(t_{i-1}, y(t_{i-1}))+4f(t_i, y(t_i))+f(t_{i+1}, y(t_{i+1}))]\]Rearranging gives Milne's formula as\[y(t_{i+1}) \approx y(t_{i-3})+\frac{h}{3}[f(t_{i-3}, y(t_{i-3}))+4f(t_{i-2}, y(t_{i-2}))+2f(t_{i-1}, y(t_{i-1}))+4f(t_i, y(t_i))+f(t_{i+1}, y(t_{i+1}))].\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Numerical Integration
Numerical integration is an essential technique in mathematics, commonly used to estimate the integral of a function when an analytic solution is difficult or impossible to find. These methods are a great help in approximating areas under curves. Unlike the traditional analytical methods, numerical integration doesn't require a simple antiderivative.

The beauty of numerical integration lies in its ability to handle complex functions over defined intervals. It breaks the interval into smaller sub-intervals and uses simple function values at these points to construct approximate solutions. This leads to estimates that can be made increasingly accurate by refining the intervals.

One popular approach for numerical integration is the Newton-Cotes formula, which uses polynomial equations to approximate the underlying integral. Each method within this formula varies in its degree of accuracy and computational effort, providing flexibility for many different kinds of problems.
  • Functions may not easily have antiderivatives.
  • Complex integrals become manageable with approximation.
  • Important in fields such as physics, engineering, and computer science.
Newton-Cotes Formula
The Newton-Cotes formula is a common method in numerical integration that approximates integrals using equidistant points. It simplifies the integral of a continuous function, making it practical when dealing with complex or non-standard functions. The Newton-Cotes methods are classified by the degree of the polynomial used to approximate the integral over an interval.

For example, the Trapezoidal Rule and Simpson's Rule are well-known Newton-Cotes methods. The formula can be either open or closed based on how points are chosen in the interval. Open Newton-Cotes does not use the endpoints of the interval, while closed versions include them.

The 4th order formula, as used in the derivation of Milne's Method, is a type of open Newton-Cotes formula. It uses five points, maintaining balance between computational efficiency and approximation accuracy.
  • Provides both open and closed methods.
  • Utilizes polynomial approximations for function fit.
  • Adjustable degree for different levels of accuracy.
Ordinary Differential Equations
Ordinary Differential Equations (ODEs) are equations involving a function and its derivatives. These equations model various real-life phenomena such as population growth, heat conduction, and mechanical vibrations. Solving ODEs means finding a function or a set of functions that satisfy the equation.

In many cases, analytic solutions of ODEs are complicated and not straightforward to obtain. This challenge is where numerical methods, like Milne's Method derived using the Newton-Cotes formula, become particularly powerful. These numerical approaches allow us to find approximate solutions over given intervals.

Milne's method, in particular, leverages these principles by using numerical techniques to predict future values of an evolving system. This is essential in fields that require precise results but face complex differential equations.
  • Widely applicable in science and engineering.
  • Solving involves finding functions that fit derivatives.
  • Numerical solutions provide feasible approximations for complex situations.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Picard's method for solving the initial-value problem $$ y^{\prime}=f(t, y), \quad a \leq t \leq b, \quad y(a)=\alpha $$ is described as follows: Let \(y_{0}(t)=\alpha\) for each \(t\) in \([a, b]\). Define a sequence \(\left\\{y_{k}(t)\right\\}\) of functions by $$ y_{k}(t)=\alpha+\int_{a}^{t} f\left(\tau, y_{k-1}(\tau)\right) d \tau, \quad k=1,2, \ldots $$ a. Integrate \(y^{\prime}=f(t, y(t))\), and use the initial condition to derive Picard's method. b. Generate \(y_{0}(t), y_{1}(t), y_{2}(t)\), and \(y_{3}(t)\) for the initial- value problem $$ y^{\prime}=-y+t+1, \quad 0 \leq t \leq 1, \quad y(0)=1 $$ c. Compare the result in part (b) to the Maclaurin series of the actual solution \(y(t)=t+e^{-t}\).

Given the initial-value problem $$ y^{\prime}=\frac{2}{t} y+t^{2} e^{t}, \quad 1 \leq t \leq 2, \quad y(1)=0 $$ with exact solution \(y(t)=t^{2}\left(e^{t}-e\right):\) a. Use Euler's method with \(h=0.1\) to approximate the solution, and compare it with the actual values of \(y\). b. Use the answers generated in part (a) and linear interpolation to approximate the following values of \(y\), and compare them to the actual values. i. \(\quad y(1.04)\) ii. \(\quad y(1.55)\) iii. \(\quad y(1.97)\) c. Compute the value of \(h\) necessary for \(\left|y\left(t_{i}\right)-w_{i}\right| \leq 0.1\), using Eq. (5.10).

Show that Heun's method can be expressed in difference form, similar to that of the Runge-Kutta method of order four, as $$ \begin{aligned} w_{0} &=\alpha \\ k_{1} &=h f\left(t_{i}, w_{i}\right), \\ k_{2} &=h f\left(t_{i}+\frac{h}{3}, w_{i}+\frac{1}{3} k_{1}\right) \\ k_{3} &=h f\left(t_{i}+\frac{2 h}{3}, w_{i}+\frac{2}{3} k_{2}\right) \\ w_{i+1} &=w_{i}+\frac{1}{4}\left(k_{1}+3 k_{3}\right) \end{aligned} $$ for each \(i=0,1, \ldots, N-1\).

A projectile of mass \(m=0.11 \mathrm{~kg}\) shot vertically upward with initial velocity \(v(0)=8 \mathrm{~m} / \mathrm{s}\) is slowed due to the force of gravity, \(F_{g}=-m g\), and due to air resistance, \(F_{r}=-k v|v|\), where \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\) and \(k=0.002 \mathrm{~kg} / \mathrm{m}\). The differential equation for the velocity \(v\) is given by $$ m v^{\prime}=-m g-k v|v| $$ a. Find the velocity after \(0.1,0.2, \ldots, 1.0 \mathrm{~s}\). b. To the nearest tenth of a second, determine when the projectile reaches its maximum height and begins falling.

Use Taylor's method of order two to approximate the solution for each of the following initial-value problems. a. \(\quad y^{\prime}=\frac{2-2 t y}{t^{2}+1}, \quad 0 \leq t \leq 1, \quad y(0)=1\), with \(h=0.1\) b. \(\quad y^{\prime}=\frac{y^{2}}{1+t}, \quad 1 \leq t \leq 2, \quad y(1)=-(\ln 2)^{-1}\), with \(h=0.1\) c. \(\quad y^{\prime}=\left(y^{2}+y\right) / t, \quad 1 \leq t \leq 3, \quad y(1)=-2\), with \(h=0.2\) d. \(\quad y^{\prime}=-t y+4 t / y, \quad 0 \leq t \leq 1, \quad y(0)=1\), with \(h=0.1\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.